simulation of wave and current forces on template offshore

SIMULATION OF WAVE AND CURRENT FORCES ON

TEMPLATE OFFSHORE STRUCTURES

Jamaloddin Noorzaei1, Samsul Imran Bahrom1*, Mohammad Saleh Jaafar1,Waleed Abdul Malik Thanoon1 and Shahrin Mohammad2

Received: Feb 22, 2005; Revised: Jun 2, 2005; Accepted: Jul 11, 2005

Abstract

This paper describes the analytical and numerical methods adopted in developing a program for

modeling wave and current forces on slender offshore structural members. Two common wave

theories have been implemented in the present study, namely Airy,s linear theory and Stokes

, fifth

order theory, based on their attractiveness for engineering use. The program is able to consider wind

drift and tidal currents by simply adding the current velocity to the water velocity caused by the waves.

The Morison equation was used for converting the velocity and acceleration terms into resultant forces and was

extended to consider arbitrary orientations of the structural members. Furthermore, this program has

been coupled to a 3-D finite element code, which can analyze any offshore structure consisting of

slender members. For calibration and for comparison purposes, the developed programs were checked against

a commercial software package called Structural Analysis Computer System (SACS). From the

simulations of wave loading and structural analysis on few model tests, it can be concluded that the

developed programs are able to reproduce results from the model tests with satisfactory accuracy.

Keywords: Offshore structures, wave and current forces, Airy,s linear theory, Stokes

, fifth order theory,

Morison equation, computer program

Introduction

It is essential for all offshore structural analysts

to estimate the forces generated by fluid loading

given the description of the wave and current

environment (Borthwick and Herbert, 1988). In

considering wave forces, the sea comprises of

a large number of periodic wave components

with different wave heights, periods and directions

of travel which all occur at the same time in a

given area. The superposition of all of these wave

componentscoupled with their dispersive

behavior leads to a randomly varying sea

surface elevation, which can be treated by

statistical methods. However, to provide

engineering solutions, the use of regular

wave theories is common, since regular wave

theories yield good mathematical models of long

crested periodic waves, which are components

of an irregular sea (Witz et al., 1994). There is

a wide range of regular wave theories ranging

from the simple Airy,s linear theory to the higher

order formulations.

In Le Mehaute et al. (1968) measured water

1 Civil Engineering Department, Faculty of Engineering, University Putra Malaysia, 43400, Serdang, Selangor,

Malaysia, tel: +603-89466371, E-mail: [email protected] Faculty of Civil Engineering, University Technology Malaysia, 81310, Johor Bahru, Malaysia* Corresponding author

suranaree J. Sci. Technol. 12(3):193-210

194 Simulation Of Wave and Current Frces on Template Offshore Strucetures

y

xH

L

d

y = 0

S, W, L, y =d

c

η

particle velocity accuracies in percentage at the

seabed, still water level, water surface as well

as overall for various regular wave theories

(Patel, 1989). From that study, both Airy,s linear

and Stokes, fifth order theory offered sufficiently

good agreement for engineering use. Witz et al.

(1994) also noted that the solution to the Stokes,

fifth order theory presented by Skjelbreia and

Hendrickson (1961) has been implemented

widely in computer programs used within the

offshore industry. Based on this, Airy,s linear

and Stokes, fifth order theories have been

implemented in the present study.

The primary objectives of the present study

is to (i) write a computer program that is able to

simulate wave and current forces on template

offshore structures using traditional numerical

methods with minimal sacrifice towards accuracy.

(ii) To couple the written program to an existing

3-D finite element program. Finally, to show the

applicability of the coupled program by analyzing

a simple offshore structure.

Wave Theories

It is necessary to define the coordinate system

and the terminology that will be used in the

development of the wave theories in this paper.

Figure 1 shows the coordinate system with x

measured in the direction of the wave propagation,

y measured upwards from the ground surface and

z orthogonal to x and y. It is assumed that the

waves are two dimensional in the x-y plane and

that they propagate over a smooth horizontal bed

in water of constant undisturbed path. Here the

following definitions hold:

SWL = still water level

d = distance from the seabed to the SWL

h = instantaneous vertical displacement

of the sea surface above the SWL

H = height of a wave

L = wavelength (usually unknown)

T = wave period (usually known)

c = speed of wave propagation (phase

speed, phase velocity, celerity, = L/T = ω/k)

k = wavenumber (=2p/L)

f = wave frequency (=1/T)

w = wave angular frequency (=2p/T =2pf )

Formulation of Airy,s Linear Theory

A relatively simple theory of wave motion,

known as Airy,s linear theory, was given by G.B. Airy

in 1842 (Dawson, 1983). This description assumes

a sinusoidal wave form whose height is small in

comparison with the wavelength and the water

depth. Although not strictly applicable to typical

design waves used in offshore structural engineering,

this theory is valuable for preliminary calculations

and for revealing the basic characteristics of

wave-induced water motion (Dawson, 1983).

Airy,s linear theory provides an expression

for horizontal and vertical water particle velocity

at place (x, y) and time, t as (Dawson, 1983):

uH ky

kdkx t= −

ωω

2coshsinh

cos( ) (1)

vH ky

kdkx t= −

ωω

2sinhsinh

sin( ) (2)

The wavenumber, k and wave angular

frequency, ω are related through the Airy,s

linear theory by the dispersion equation:

Figure 1. Definition sketch for progressive waves

195Suranaree J. Sci. Technol. Vol. 13 No. 3; July-September 2005

ω 2 = gk kdtanh (3)

Using the dispersion equation above, the

wave speed may be expressed as:

cg

kkd= ( tanh ) /1 2

(4)

The water particle accelerations are

obtained as: a du dt a dv dtx y≈ ≈/ /and , so

that in using Eqns. (1) and (2):

aH ky

kdkx tx = −

ωω

2

2coshsinh

sin( ) (5)

aH ky

kdkx ty = −

ωω

2

2sinhsinh

cos( ) (6)

The inherent assumption in the derivation

of Airy,s linear theory has a limit of y = d, which

does not allow computation above the SWL (i.e.

y > d). This predicament is resolved by the linear

surface correction, η (Charkrabarti, 1990):

η ω= −H

kx t2

cos( ) (7)

Thus, at the free water surface, the vertical

position of the wave becomes:

y d= +η (8)

Formulation of Stokes, Fifth Order

Theory

Stokes, fifth order theory is derived by substituting

Taylor series approximations for the variables

in the free surface boundary conditions; the

order of solution depends on the number of

Taylor series terms included (Williams et al.,

1998). The method of solution for the Stokes,

fifth order theory adopted in this paper is based

on the methods suggested by Skjelbreia and

Hendrickson (1961). Most of the algebraic

complexities in their solution are in the coefficients,

A denoting wave velocity parameters, B denoting

wave-profile parameters and C denoting

frequency parameters. These coefficients are

given in explicit form by Skjelbreia and

Hendrickson (1961).

The instantaneous vertical displacement of

sea surface above the SWL according to Stokes,

fifth order theory is described as (Dawson, 1983):

η ω= −=

∑1

1

5

kF n kx tn

n

cos ( ) (9)

where the coefficients, Fn are given in terms

of λ and B (refer Appendix). λ denotes a

wave-height parameter.

For a design wave, λ and k are to be

determined by virtue of the following pair of

equations (Sarpkaya and Isaacson, 1981):

12

333

535 55kd

B B BH

dλ λ λ+ + +[ ] =( ) (10)

and

kd kd C Cd

gTtanh( ) 1 42

14

22

2+ +[ ] =λ λ π (11)

Bhattacharya (1991) describes a solution

to the above equations using the Newton

Rhapson method. Once the values of k and λare found, the solution will then be complete

and the remaining variables of interest may

readily be evaluated.

The horizontal water velocity and the

vertical water velocity are expressible as:

uk

Gnky

nkdn kx tn

n

= −=

∑ωω

coshsinh

cos ( )1

5

(12)

vk

Gnky

nkdn kx tn

n

= −=

∑ωω

1

5 sinhsinh

sin ( ) (13)

where the coefficients, Gn are functions of A

(refer Appendix).

In addition to the previous relations, it is

also necessary to have the frequency relation

connecting the wave angular frequency, ωwith

the wavenumber, k. This relation is given by the

equation (Dawson, 1983):


y

z

x

WnUnx

UnzUny

ω 2 21

421= + +gk a C a C kd( )tanh (14)

The wave speed is determined as in Airy,s

linear theory from the relation c = ω/k, which

for the Stokes, fifth order theory is expressible as:

cg

ka C a C kd= + +

⎡

⎣⎢⎤

⎦⎥( ) tanh

/

1 21

42

1 2

(15)

The horizontal acceleration and vertical

acceleration of the water particles can be

determined respectively from the equations:

au

tuu

xvu

yx =∂∂

+∂∂

+∂∂

(16)

av

tuv

xvv

yy =∂∂

+∂∂

+∂∂

(17)

or can be written in the following explicit forms

akc

R n kx tx n

n

= −=

∑2

1

5

2sin ( )ω (18)

akc

S n kx ty n

n

=−

−=

∑2

1

5

2cos ( )ω (19)

where the coefficients, Rn and Sn are given in

terms of Un and Vn (refer Appendix) :

U Gnky

nkdn n=coshsinh

(20)

V Gnky

nkdn n=sinhsinh

(21)

Determination of Wave Forces

For slender offshore structures such as monopiles,

tripods or template offshore structures, the

Morison equation is used for converting the

velocity and acceleration terms into wave forces

(Henderson et al., 2003). The Morison equation

maybe expressed as:

f C D uu CD

aD x= +π1

2 41

2

ρ ρ (22)

Where ρ denotes water density, CD and CI

denote the drag and inertia coefficients

respectively and D is the diameter of the member.

The first term on the right hand side of this

equation is referred to as the drag term and is

proportional to the square of the water velocity.

The second term is referred as the inertia term

and is proportional to the water acceleration.

The most important consideration in

applying Morison,s equation is the selection of

appropriate drag and inertia coefficients.

However, there is considerable uncertainty in the

CD and CI values appropriate for the calculation

of offshore structural members, with many

values in publication. Cassidy (1999) reviewed

some published studies in the literature. He found

that CD ranged from 0.6 for smooth cylinders to

1.2 for rough cylinders. CI ranged from 1.75 for

rough cylinders to 2.0 for smooth cylinders.

The values of u and ax in the Morison

equation are calculated from an appropriate

wave theory, together with chosen values of CD

and CI. Eqn. (22) yields at any instant in the wave

cycle, the force distribution along the member.

Wave Forces on Arbitrarily Oriented

Cylinders

The direction of wave force normal to the

cylinder may conveniently be resolved into

horizontal and vertical components. To illustrate,

consider a fixed cylinder arbitrarily inclined to

axes x, y and z as shown in Figure 2.

Figure 2. Definition sketch for an inclined

cylinder (After Sarpkaya and

Isaacson, 1981)


With polar coordinates φ and ψ defining

the orientation of the cylinder axis, the magnitude,

Wn of the water velocity normal to the cylinder

axis is given by:

W u v c u c vn x y= + − +[ ( ) ] /2 2 2 1 2 (23)

and its components in the x, y, and z directions

are given respectively by:

U u c c u c y

U v c c u c y

U c c u c y

nx x x y

ny y x y

nz z x y

= − +

= − +

= − +

( )

( )

( )

(24)

where,

c c

cx y

z

= =

=

sin cos , cos ,

sin sin

φ ψ φ

φ ψ(25a:b:c)

The components of the water acceleration

in the x, y, and z directions are given, respec-

tively by:

a a c c a c a

a a c c a c a

a c c a c a

nx x x x x y y

nx y y x x y y

nz z x x y y

= − +

= − +

= +

( )

( )

( )(26)

With these relations, the components of

the force per unit of cylinder length acting in

the x, y, and z directions are given respectively

by the generalized Morison equations:

f

f

f

C DW

U

U

U

C D

a

a

a

x

y

z

D n

nx

ny

nz

nx

ny

nz

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

=

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

+ π

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

0 5 0 25 1. .ρ ρ (27)

A typical offshore structural beam element

may be subjected to non-uniformly distributed

loading along its length arising from the above

equation. These can readily be translated into

forces at the beam fixed end using equilibrium

equations (Witz et al., 1994).

The total forces are calculated by numerical

integration of the relations:

F f ds F f ds

F f ds

x xs

y ys

z zs

= =

=

∫ ∫

∫

, ,

(28a:b:c)

where s denotes the distance along the member

axis, and the limits on the integrals are chosen

to include all of the member on which the wave

force acts.

Current Velocity

The most common currents considered in

offshore structural analysis are tidal currents and

wind drift currents (Dawson, 1983). Both of

these currents are usually regarded as horizontal

and varying with depth.

The tidal current velocity profile at any

vertical distance from the seabed may be

determined as (Dawson, 1983):

U y Uy

dT oT( )/

=⎛

⎝⎜

⎞

⎠⎟

1 7

(29)

and, the wind drift current velocity profile may

be determined as:

U y Uy

dw ow( ) =⎛

⎝⎜

⎞

⎠⎟ (30)

where, d denotes the water depth, y is the

vertical distance from the seabed, UoT and UoW

denote the tidal and wind drift current velocity

at the water surface respectively.

For regular design waves and a horizontal

current of arbitrary depth variation, the force

exerted on an offshore structure is normally

calculated by simply adding the horizontal

water velocity caused by the waves to that

component of current velocity (Dawson, 1983).

Verification of the Computer Program

In previous articles, brief discussion on the

theoretical aspect and simulation of the wave

forces on offshore structural members has been

presented. A computer program written in the

FORTRAN language working under the

Microsoft Power Station environment has been

written. The program has been validated with a

standard commercial package called Structural


Analysis Computer System (SACS, version 5.1,

2001).

SACS represent wave loads that have

a curved or non-linear distribution by a series of

linear varying load segments using a curve

fitting technique. Velocity and acceleration

values are calculated for each end of the member

and a linear variation is assumed between the

ends. The velocity and acceleration values at the

member center are calculated and compared to

the values predicted by the linear variation.

If either is more than 5% different from

the linear distribution, then the member is

segmented to include the centre point of the

member. Themember would now have two

linear load segments. This is repeated until the

5% criterion is met. The user also has the option

to set the number of equal segments desired.

However, the SACS program is limited to a

maximum of 10 segments (SACS Users Manual,

2001).

In the present study, the total forces are

also calculated with linear segments, but

without the curve fitting technique. Valuable

programming time could be saved if the error

committed by using a fixed number of segments

compared to auto segmentation is small.

Nevertheless, the present study it is not limited

to 10 segments. We are to see the effects of the

different number of segments with respect to the

results of SACS auto segmentation. Figure

3(a) shows a discreteness of load segments of

the present study for ten segments while Figure

3(b) shows a possible discreteness of load

segments on a member by SACS for ten segments.

The written wave simulation program has

been attached to a 3-D finite element program

and the new version of the coupled finite

element program is validated by analyzing

a simple offshore structure by comparing the

results obtained by the present study to the SACS

commercial program.

Numerical Examples

For the purposes of calibration and

comparison, three numerical examples have been

selected, namely:

• Numerical Example I - (comparing the

results of total forces of the present study

to that of SACS for a vertical cylinder).

• Numerical Example II - (comparing the

results of total forces of the present study

to that of SACS for an inclined cylinder).

• Numerical Example III - (structural

analysis of a simple offshore structure).

Numerical Example I and II were tested

under the following cases:

• Case I - Airy,s linear theory

• Case II - Stokes, fifth order theory

Numerical Example I - Problem

Definition

In this example, the cylinder is considered to be

in the vertical position. Initially for Case I, the

forces arising from Airy,s linear theory would

Figure 3. (a) Discreteness of the present study for ten segments

(b) A possible discreteness on a member by SACS for ten segments

(a)

_ _ _ _ _

_ _ _

_ _ _ _ _ _ _ _ _ _ _ _

L/10

L/10

L/10

L/10

L/10 L/10 L/10

L/10

L/10

L/10

1

2

3

4

5

6

7

8

10

9

(b)

_ _ _ _ _

_ _ _

_ _ _ _ _ _ _ _ _ _ _ _

L

1

34

67

8

9

2

5

10


be calculated and subsequently for Case II,

the forces arising from Stokes, fifth order

theory would be calculated. In each Case, the

distributed wave force acting on the cylinder

arising the from present study would divided into

5, 10 and 15 segments respectively to calibrate

and compare which number of segments would

correspond closest to the results of the SACS

program. The wave parameters and cylinder

details used in Numerical Example I are

presented in Figure 4. The values of CD and CI

are based from Dawson (1983).

Results and Discussion on Numerical

Examble I

Case I

Figure 5 shows the distribution of wave

forces plus currents for a vertical cylinder

arising from Airy,s linear theory for different

values of phase angle. The data in that figure

show that all results of the present study slightly

underestimated the results of the SACS program.

The average percentage error of the present study

compared to SACS is 1.68%, 2.62%, and 2.80%

for 5, 10, and 15 segments respectively. The

slight disagreement between the present study

to that of SACS may lie in the tolerance for the

iteration of Egn. (3) to obtain the wavenumber,

k. In the present study, the tolerance was set to 5

decimal places. From Table 1, we can see a slight

difference for the wavenumber value obtained

Figure 4. Definition sketch for numerical example I

from the present study to that of SACS. The

wavenumber is used in most equations of the

wave kinematics, thus affecting subsequent

results. Another evident reason for the disagreements

is of course, the auto segmentation of the SACS

program. The free water surface profile predicted

by present study to that of SACS are plotted in

Figure 6. It is seen from these plots, both

programs gave identical results.

As mentioned earlier, Airy,s linear theory

is not generally valid for deep water, thus the

estimated error in using this theory over the more

accurate theories can be made by utilizing

Figure 7 (Dawson, 1983). It is obvious from this

figure that the ratio of water depth to wavelength

for this case is approximately 2.0. But the ratio

of water depth to wavelength of 0.2 occurs at

a value of the ratio of wave height to wavelength

of 0.04. This corresponds to a 10% error. For

the design wave in this example, the ratio of

wave height to wavelength is 0.09, thus

the estimated error in using Airy,s linear theory

is approximately 10 x 0.09 / 0.04 = 22%.

Case II

Figure 8 shows the distribution of wave

forces plus currents for a vertical cylinder

arising from Stokes, fifth order theory for

different phase angles. A similar trend can be

seen to that of the results of the previous case,

with all results of the present study slightly

underestimating the results of the SACS

k

jNote: Note to Scale


Program Wavenumber Wavelength (m)

Present study 0.05503 114.168

SACS 0.05506 114.105

program. The average percentage difference of

the present study compared to SACS is 1.96%,

2.62%, and 2.74%, for 5, 10, and 15 segments

respectively.

Figure 9 shows the surface elevation

arising from the Stokes, fifth order theory for

different phase angles. It can be seen that the

results of SACS and the present study have good

agreement in modeling the free water surface.

Table 2 shows the wavenumber and wavelength

predicted by SACS and the present study arising

from Stokes, fifth order theory. From that table,

it is seen that the present study underestimated

the wavelength by 0.64 m. This may be due to

the techniquesemployed in solving Eqns. 10 and 11.

Even with the same wave parameters as

in Case I, it is clear that in this Case, wave loads

predicted from Stokes, fifth order theory are

significantly higher compared to that of Airy,s

linear theory. From Figures 5 and 8 the maximum

horizontal force arising from Airy,s linear theory

predicted by SACS is 357.46 KN and the

maximum horizontal force arising form Stokes,

fifth order theory predicted by SACS is 444.35

KN respectively. In this Case, Airy,s linear

theory underestimated the forces arising from

Stokes, fifth order theory by 19.6%, which very

close to the estimated error discussed in Case I.

Figure 5. Distribution of wave forces plus currents for a vertical cylinder arising from

Airy,s linear theory

Table 1. Wavenumber and wavelength predicted by SACS and the present study arising

from Airyû s linear theory

Among the reasons why Stokes, fifth

order theory predicts larger values of forces in

this case are: (i) Stokes, fifth order theory

models the free water surface more accurately

producing steeper crests and lower troughs as

illustrated in Figure 9. In this Case, the crests

obtained using Stokes, fifth order theory

increased by about 1.52 m and the through

increased by about 1.49 m compared to Airy,s

linear theory. (ii) Stokes, fifth order theory

retains the convective accelerations terms

(Egns. 16 and 17) whereas Airy,s linear theory

neglects them.

Numerical Example II - ProblemDefinition

In this example, the validation procedures,

assumptions and wave parameters made are

similar to Numerical Example I, the exception

being that in this case, the cylinder is considered

to be in an arbitrarily oriented position. It is

important to test the cylinder in an arbitrarily

oriented position as to consider the forces in

the y and z directions and the geometrical

effect of the oriented member towards the

free water surface profile. A definition sketch

for Numerical Example II is illustrated in

Figure 10.

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- - -

-

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Dis

tane

fre

m S

WL

(m

) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0 50 100 150


200 250 300 350

6

4

2

0

-2

-4

-6

Present study SACS

Wav

e he

ight

Wav

e he

ight

Wave heightWave height

Cno

idal

theo

ry

Stokes,fifth order theory

Airy,s linear theory

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.12

0.10

0.08

0.06

0.04

0.02

0

Figure 7. Diagram showing the range of validity of Airy,s linear theory, assuming tolerable

errors of no more than 10%

Figure 6. Surface elevation arising from Airy,s linear theory at different phase angles

Results and Discussion on Numerical

Example-II

Case I

Figure 11(a) shows the distribution

of wave forces plus currents for an inclined

cylinder arising from Airy,s linear theory for

different phase angles in the x-direction. For

forces in the x-direction, there is a good agreement

between the distributions of the forces predicted

by both programs, however in the phase angle

between 235 and 250 degrees there is a slight

deviation between the two programs. Due to the

scale chosen, these values are not apparent in

Figure 11(a). This figure is magnified in Figure

12 to illustrate errors obtained in this Case. In

this figure for example, the force obtained by

SACS at the phase angle of 235 degrees is -1.17

KN, and at the same phase angle, the present

study (for 5 segments) obtained a value of -2.84 KN.

The distributions of forces in the y and z

directions for different phase angles are

illustrated in Figures 11(b-c) respectively. It is

clear from these plots, that there is a good

agreement between the forces evaluated from

the present study to that of SACS.

The surface elevation with respect to

the orientation of the member obtained by

the present study is presented in Figure 13.

It can be seen that because the member

is arbitrarily oriented, the crest is produced

later compared to the vertical condition.

The crest of the water surface for the oriented

member in this Case occurs at approximately

15.32 m from the origin.

Case II

Figures 14(a-c) show the distribution of wave

forces plus currents for an inclined cylinder

arising from Stokes, fifth order theory for different

phase angles in the x, y and z directions

respectively. It is obvious from these plots that

there is a good comparison between the results


-- -

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--

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- -

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- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Fx (

KN

)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

500

400

300

200

100

0

0 50 100 150


200 250 300-100

-200

SACS (auto segmentation) Present Study (5 Segments)

Present Study (10 Segments) Present Study (15 Segments) �

� ��

�

��

�

�

�

�

�

�

�

��

� ��

� � ��

� � � ��

��

��

�

�

�

�

�

�

�

�

�

350

Figure 8. Distribution of wave forces plus currents for a vertical cylinder arising from

Stokes, fifth order theory

Figure 9. Surface elevation arising from Stokes, fifth order theory at different phase angles

predicted by both programs. However,

Figure 14(c) indicates that there is a marginal

difference for the forces between the phase angle

of 300 and 360 degrees. This could be due to

the large number and complex nature of the

coefficients for higher order solutions, thus small

algebraic errors can occur. Results reported by

different analysts tend to vary as reported in the

literature by Barltrop and Adams (1991).

Figure 15 compares the surface elevation

for the previous vertical position of the member

(Numerical example I, Case II) to the present

inclined position. The distance from origin where

the crest occurs for this case was evaluated

approximately 16.08 m from the origin.

Numerical Example III - Problem

Definition

The accuracy of the simulation program

to calculate wave and current forces has

been presented in the previous two examples.

In the next phase of this investigation,

this program is coupled to an existing 3-D

general purpose finite element code. The

verification of the coupled program is shown

by analyzing a simple offshore structure.

The formulations of the finite element program

is beyond this paper, however conventional

beam elements have been used that closely

follows the formulation presented Figure 16

shows the 3-D finite element idealization of

a simple offshore structure (Dawson, 1983)

used in this example, with the wave propagating

in the x-direction.

Results for Numerical Example III

The position of the wave crest in relation to the

structure is an important parameter, which must

be inputted into the computer program. The wave

crest should be located such that the maximum

possible shear and moments are applied to the

structure (McClelland and Reifel, 1986). Figures

17 and 18 shows the phase angle that causes the

maximum horizontal force (resulting in maximum

shear) arising from Airy,s linear and Stokes, fifth

order theory respectively. From these two figures,

the total horizontal force arising from Airy,s

linear theory and Stokes, fifth order theory are

- - -

- - -

- - -

- - -

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- - -

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- -

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-

- -

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-

- -

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- - -

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- - -

-

- -

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- - -

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- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

-

Dis

tane

fre

m S

WL

(m

)

0 50 100 150


200 250 300 350

Present study SACS

8

6

4

2

0

-2

-4

-6


Figure 10. Definition sketch for numerical example II

Table 2. Wavenumber and wavelength predicted by SACS and the present study arising

from Stokes, fifth order theory

Program Wavenumber Wavelength (m)

Present study 0.05067 124.002

SACS 0.05039 124.692

y

xzk

j

Wave Parameters:Wave period, T = 9.27 secWaveheight, H = 10.660 mWater Depth, d = 22.860 mDrage Coefficient, CD = 1Inertia Coefficient CI = 2Wind Drift Current = 1.5 m/sec

Cylinder Properties:Diameter, D = 1.2192 mposition (x, y, z):j (0.0, 0.0, 0.0)k (17.91, 33.04, 6.51)

Note: Note to Scale

almost identical. The phase angle that causes the

maximum horizontal force coincidently occurs

at a phase angle of 0 and 360 degrees. Thus, any

one of these values may be inputted in the

program since 0 degrees and 360 degrees are

actually the same position in a wave cycle.

Figure 19 shows the displacements along

leg A for forces arising from Airy,s linear theory.

From Figure 19, the present study obtained

smaller values with respect to SACS auto

segmentation with a percentage difference for

node 3 of 5.43%, 5.71%, and 5.76%, for 5, 10,

and 15 segments respectively. It can be stated

that as number of segments are increasing

due to the load redistribution, the resulting

displacements are converging. However, an

attempt has been made to divide the load

distribution into 10 equal segments for both

programs in order to have a clear comparison

between them. Thus, the displacements due

to SACS set for 10 equal segments are also

exhibited in Figure 19. In this case, the percentage

difference between SACS set for 10 equal seg-

ments to the present study set for 10 equal seg-

ments is only 1.56%.

Figure 20 shows the displacements along

leg A for forces arising from Stokesí fifth order

theory. From Figure 20, the present study

obtained smaller values with respect to SACS

auto segmentation with a percentage difference

for node 3 of 6.1%, 6.41%, and 6.46% for 5, 10,

and 15 segments respectively. The percentage

difference between SACS set for 10 equal

segments to the present study set for 10 equal

segments in this case is 2.30%.

It is established that SACS tend to produce

larger values compared to the present study when

the auto segmentation option is used. The deflected

profile for the entire offshore structure obtained

from the present study is illustrated in Figure 21.

The comparison of member end forces and

moments for selective elements obtained through

the present study and SACS (using Airy,s linear

theory and Stokes, fifth order theory) are

tabulated in Tables 3 through 6 respectively.

It is clear from these tables, that the coupled

program is able to reproduce results with respect

to SACS with good accuracy.


- -------

Fx

(KN

)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

25

20

15

10

5

0

-5 Phase Angle (degrees)

22.69

20.466

14.59

12.523

4.764

235 237 239 241 243 245 247 249 -1.17-2.841

6.61

SACS (auto segmentation)

Present Study (5 Segments)

Figure 11. Comparison of the distribution of wave forces plus currents obtained

from the present study and SACS arising from Airy,s linear theory at different

phase angles in x, y and z directions respectively

Figure 12. Magnification of Figure 11(a) for phase angle between 235 and 250 degrees

a. Force in x -direction

-150

-100

-50

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350


)N

K( xF

SACS (auto segmentation) Present Study (5 segments)

Present Study (10 Segments) Present Study (15 Segments)

b. Force in y -direction

-200

-150

-100

-50

0

50

100

0 50 100 150 200 250 300 350


)N

K( yF



c. Force in z -direction

-25-20

-15-10

-50

510

1520

2530

0 50 100 150 200 250 300 350Phase Angle (degrees)

)N

K( zF




- - -

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-

- -

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-

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Dis

tane

fre

m S

WL

(m

)

0 20 40 60 80 100

6

4

2

0

-2

-4

-6

Vertical Cylinder of Case I Inclinder Cyinder of Case -II

Figure 13. Comparison of the surface elevation arising from Airyûs linear theory for

a vertical and inclined member

a. Force in x -direction

-100

-50

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350Phase Angle (degrees)

)N

K( xF

SACS (auto segmentation) 5 Segments10 segments 15 Segments

b. Force in y -direction

-200

-150

-100

-50

0

50

100

0 50 100 150 200 250 300 350


)N

K( yF

SACS (auto segmentation) 5 Segments

10 Segments 15 Segments

c. Force in z -direction

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 50 100 150 200 250 300 350


)N

K( zF

SACS (auto segmentation) 5 Segments10 Segments 15 Segments

Figure 14. Comparison of the distribution of wave forces plus currents obtained

from the present study and SACS arising from Stokes, fifth order theory

at different phase angles in x, y, and z directions respectively


600

500

400

0 50 100 150 200 250 300 350

300

200

100

0

-100

-200

-300

Figure 17. The phase angle resulting in maximum horizontal force arising from Airyûslinear theory

Figure 15. Comparison of surface elevation arising from stokes, fifth order theory for

a vertical and inclined member

Figure 16. Offshore structure considered in numerical example III (After Dawson, 1983)

- - -

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-

- -

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-

- -

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- -

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- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

- - -

-

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Dis

tane

fre

m S

WL

(m

)

Distane frem x-axis (m)

Vertical Cylinder of Case I Inclinder Cyinder of Case -II

8

6

4

2

0

-2

-4

-6

0 20 40 60 80 100 120


-300

-200

-100

0

100

200

300

400

500

600

0 50 100 150 200 250 300 350

Phase angle (degrees)

)N

K( xF

SACS Present Study

0

5

10

15

20

25

30

35

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

Displacement (cm)

m)

( debaes morf ecnatsi

D

SACS (Auto Segmentation) 5 Segments


SACS (10 Segments)

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Figure 18. The phase angle resulting in maximum horizontal force arising from Stokes,

fifth order theory

Figure 20. Comparison of displacements of the present study vs SACS resulting from Stokes,

fifth order theory

0

5

10

15

20

25

30

35

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

Displacement (cm)

m)

( debaes morf ecnatsi

D

SACS (Auto Segmentation) 5 Segments


SACS (10 Segments)

_ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Figure 19. Comparison of displacements of the present study vs SACS resulting from Airy,s

linear theory


Table 4. Comparison of member end forces for selective members obtained from SACS

and the present study arising from Stokes, fifth order theory

Member Member Force (KN)

no. ends SACS (auto segmentation) Present study (5 segments)

Fx Fy Fz Fx Fy Fz

17 3 84.15 4.01 1.24 82.22 4.01 1.24

6 -84.15 -4.01 -1.24 -82.22 -4.01 -1.24

18 3 3.67 0.19 -0.81 3.65 0.19 -0.85

9 -3.67 -0.19 0.81 -3.65 -0.19 0.85

19 6 99.65 4.11 -3.25 97.59 4.11 -3.16

12 -99.65 -4.11 3.25 -97.59 -4.11 3.16

20 9 -3.17 -0.20 -0.19 -2.99 -0.18 -0.22

12 3.17 0.20 0.19 2.99 0.18 0.22

Member MemberForce (KN)


Fx Fy Fz Fx Fy Fz

17 3 84.15 4.01 1.24 82.22 4.01 1.24

6 -84.15 -4.01 -1.24 -82.22 -4.01 -1.24

18 3 3.67 0.19 -0.81 3.65 0.19 -0.85

9 -3.67 -0.19 0.81 -3.65 -0.19 0.85

19 6 99.65 4.11 -3.25 97.59 4.11 -3.16

12 -99.65 -4.11 3.25 -97.59 -4.11 3.16

20 9 -3.17 -0.20 -0.19 -2.99 -0.18 -0.22

12 3.17 0.20 0.19 2.99 0.18 0.22

Table 3. Comparison of member end forces for selective members obtained from SACS

and the present study arising from Airy,s linear theory

Y

X

Z

Figure 21. Deflected profile of the structure


Conclusion

From the simulations of wave loading on

a vertical and inclined cylinder, it can be

concluded that the developed program is able to

reproduce results from the model tests with

satisfactory accuracy. The wave simulation

program has been coupled with a 3-D finite

element program and the applicability and

accuracy of the coupled program has been

demonstrated by analyzing a simple offshore

structure. For individual members, the error

committed in using Airy,s linear theory for deep

water is apparent, however when analyzing

a whole offshore structure, the theory is able to

give a good representation of the wave loads

compared to the more accurate but complex

Stokes, fifth order theory. It is seen that SACS

auto segmentation will give larger results

compared to dividing the load distribution into

equal segments. The wave characteristics

produced by the present study are also in agreement

with what is available in the literature.

Acknowledgements

The authors would like to thank Mr. Shaharuddin

Ismail of Malaysian Mining Coorporartion

(MMC) oil and gas Engineering and Ir. Rafee

Makbol (formerly of MMC oil and gas Engineering)

for making the results of the SACS program

available to the authors.



Fx Fy Fz Fx Fy Fz

17 3 0.35 -10.52 30.79 0.34 -10.50 30.85

6 -0.35 -8.36 30.29 -0.34 -8.41 30.33

18 3 0.22 -1.48 1.34 0.21 -0.90 1.36

9 -0.22 13.75 1.60 -0.21 13.91 1.59

19 6 0.45 27.35 31.95 0.43 26.63 31.96

12 -0.452 2.20 30.72 -0.43 21.53 30.73

20 9 -0.05 -2.53 -1.48 -0.06 -2.10 -1.41

12 0.05 5.39 -1.50 0.06 5.52 -1.40

Table 5. Comparison of member end moments for selective members obtained from SACS

and the present study arising from Airy,s linear theory

Table 6. Comparsison of member end moments for selective members obtained from SACS

and the present study arising from Stokesû fifth order theory



Fx Fy Fz Fx Fy Fz

17 3 0.35 -10.52 30.79 0.34 -10.50 30.85

6 -0.35 -8.36 30.29 -0.34 -8.41 30.33

18 3 0.22 -1.48 1.34 0.21 -0.90 1.36

9 -0.22 13.75 1.60 -0.21 13.91 1.59

19 6 0.45 27.35 31.95 0.43 26.63 31.96

12 -0.45 22.20 30.72 -0.43 21.53 30.73

20 9 -0.05 -2.53 -1.48 -0.06 -2.10 -1.41

12 0.05 5.39 -1.50 0.06 5.52 -1.40


References

Barltrop, N.D.P., and Adams, A.J. (1991).

Dynamics of fixed marine structures.

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Appendix

F1 = λF2 = λ2Β22

+ λ4Β24

F3 = λ3Β33 + λ5Β35 (Α.1)

F4 = λ4Β44

F5 = λ5Β55

G1 = λΑ11 sin kd +λ3 Α13 sin kd +λ5Α15 sin kd

G2 = 2λ2(Α22 sin2kd +λ4 sin 2kd)

G3 = 3(λ3 sin3kd +λ5 sin 3kd) (Α.2)G4 = 4λ4Α44

sin4kd

G5 = 5(λ5Α55 sin5kd

R1 = 2U1 -U1U2

-V1V2 -U2U3

-V2V3

R2 = 4U2 -U1

2 +V12 -2U1U3

-2V1V3

R3 = 6U3 -3U1U2

+3V1V2 -3U1U4

-3V1V4 (Α.3)R4 = 8U4

-2U22 +2V2

2 -4U1U3 +4V1V3

R3 = 10U5 -5U1U4

-5U2U3 +5V1V4

+5V2V3

S0 = -2U1V1

S1 = 2V1 -3U1V2 -3U2V1 -5U2V3-5U3V2

S2 = 4V2 -4U1V3 -4U3V1

S3 = 6V3-U1V2+U2V1- 5U4V1 - 5U4V1 (Α.4)

S4 = 8V4 -2U1V3 +2U3V1 +4U2V2

S5 = 10V5 -3U1V4 +3U4V1 -U2V3 +U3V2

simulation of wave and current forces on template offshore

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